Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Models of low-speed flow for near-critical fluids with gravitational and capillary effects

Authors: D. L. Denny and R. L. Pego
Journal: Quart. Appl. Math. 58 (2000), 103-125
MSC: Primary 76N10; Secondary 35Q35, 76D45
DOI: https://doi.org/10.1090/qam/1738560
MathSciNet review: MR1738560
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Abstract: We study low-speed flows of a highly compressible, single-phase fluid in the presence of gravity, for example, in a regime appropriate for modeling recent space-shuttle experiments on fluids near the liquid-vapor critical point. In the equations of motion, we include forces due to capillary stresses that arise from a contribution made by strong density gradients to the free energy. We derive formally simplified sets of equations in a low-speed limit analogous to the zero Mach number limit in combustion theory.

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  • [1] D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech. 30, 139-165 (1998) MR 1609626
  • [2] G. K. Batchelor, The conditions for dynamical similarity of motions of a frictionless perfect-gas atmosphere, Quart. J. Roy. Meteo. Soc. 79, 224-235 (1953)
  • [3] R. F. Berg, Thermal equilibration near the critical point: Effects due to three dimensions and gravity, Phys. Rev. E 48, 1799-1805 (1993)
  • [4] H. Boukari, M. E. Briggs, J. N. Shaumeyer, and R. W. Gammon, Critical speeding up observed, Phys. Rev. Lett. 65, 2654-2657 (1990)
  • [5] H. Boukari, R. Pego, and R. W. Gammon, Calculation of the dynamics of gravity-induced density profiles near a liquid-vapor critical point, Phys. Rev. E 52, 1614-1625 (1995)
  • [6] H. Boukari, J. N. Shaumeyer, M. E. Briggs, and R. W. Gammon, Critical speeding up in pure fluids, Phys. Rev. A 41, 2260-2263 (1990)
  • [7] A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd. ed., Springer-Verlag, New York, 1993 MR 1218879
  • [8] D. L. Denny and R. L. Pego, Solutions for a model of low-speed flow for highly compressible fluids with capillary effects, in preparation
  • [9] J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal. 88, 95-133 (1985) MR 775366
  • [10] D. Durran, Improving the anelastic approximation, J. Atmos. Sci. 46, 1453-1461 (1989)
  • [11] J. A. Dutton and G. H. Fichtl, Approximate equations of motion for gases and liquids, J. Atmos. Sci. 26, 241-254 (1969)
  • [12] P. Embid, Well-posedness of the Nonlinear Equations for Zero Mach Number Combustion, Ph. D. Thesis, University of California, Berkeley, 1984 MR 888460
  • [13] R. Gammon, personal communication. Also see the ZENO home page at the URL http://roissy.umd.edu/. The experimental design is described at the URL http://roissy. umd.edu/usmp3/reminder.html.
  • [14] C. Ikier, H. Klein, and D. Woermann, Optical observation of the gas/liquid, phase transition in near-critical $ SF_{6}$ under reduced gravity, J. Coll. Internat. Sci. 178, 368-370 (1996)
  • [15] C. Ikier, H. Klein, and D. Woermann, Density equilibration in a near-critical fluid under reduced gravity, Ber. Bunsenges. Phys. Chem. 100 (8), 1308-1311 (1996)
  • [16] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34, 481-524 (1981) MR 615627
  • [17] A. B. Kogan, F. Zhong, and H. Meyer, Dynamics of density equilibration near the liquid-vapor critical point of He--3, Czechoslovak J. Phys. 46, Suppl. 1, 71-72 (1996)
  • [18] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed., Pergamon, Oxford, 1987 MR 961259
  • [19] F. B. Lipps and R. S. Hemler, A scale analysis of deep moist convection and some related numerical calculations, J. Atmos. Sci. 39, 2192-2210 (1982)
  • [20] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53, Springer, New York, 1984 MR 748308
  • [21] A. Majda and J. Sethian, The derivation and numerical solution of the equations for zero Mach number combustion, Combust. Sci. Tech. 42, 185-205 (1985)
  • [22] M. R. Moldover, J. V. Sengers, R. W. Gammon, and J. R. Hocken, Gravity effects in fluids near the gas-liquid critical point, Rev. Mod. Phys. 51, 79-99 (1979)
  • [23] Y. Ogura and N. A. Phillips, Scale analysis of deep and shallow convection in the atmosphere, J. Atmos. Sci. 19, 173-179 (1962)
  • [24] A. Onuki and R. A. Ferrell, Adiabatic heating effect near the gas-liquid critical point, Physica A 164, 245-264 (1990)
  • [25] A. Onuki, H. Hao, and R. A. Ferrell, Fast adiabatic equilibration in a single-component fluid near the liquid-vapor critical point, Phys. Review A 41, 2256-2259 (1990)
  • [26] V. A. Rabinovich, Thermophysical Properties of Neon, Argon, Krypton, and Xenon, Hemisphere Publishing Corp., New York, 1988
  • [27] R. G. Rehm and H. R. Baum, The equations of motion for thermally driven, buoyant flows, J. Res. Natl. Bur. Stand. 83, 297-308 (1973)
  • [28] J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity, Clarendon Press, Oxford, 1982
  • [29] J. V. Sengers, Transport properties of fluids near critical points, Internat. J. Thermophysics 6, 203-232 (1985)
  • [30] J. V. Sengers, R. S. Basu, and J. M. H. Levelt Sengers, Representative equations for the thermodynamic and transport properties of fluids near the gas-liquid critical point, NASA Contractor Report 3424, 1981
  • [31] H. L. Swinney and D. L. Henry, Dynamics of fluids near the critical point: Decay rate of order-parameter fluctuations, Phys. Rev. A 8, 2586-2617 (1973), and references therein
  • [32] R. Wilhelmson and Y. Ogura, The pressure perturbation and the numerical modeling of a cloud, J. Atmos. Sci. 29, 1295-1307 (1972)
  • [33] C.-S. Yih, Stratified Flows, Academic Press, New York, 1980 MR 569474
  • [34] B. Zappoli and P. Carles, The thermo-acoustic nature of the critical speeding up, Eur. J. Mech. B/Fluids 14, 41-65 (1995)
  • [35] B. Zappoli, S. Amiroudine, P. Carlès, and J. Ouazzani, Thermoacoustic and buoyancy-driven transport in a square side heated cavity filled with a near critical fluid, J. Fluid Mech. 316, 53-72 (1996)
  • [36] F. Zhong and H. Meyer, Density equilibration near the liquid-vapor critical point of a pure fluid: Single phase, Phys. Rev. E 51 3223-3241 (1995)

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DOI: https://doi.org/10.1090/qam/1738560
Article copyright: © Copyright 2000 American Mathematical Society

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